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Theorem supsn 7931
Description: The supremum of a singleton. (Contributed by NM, 2-Oct-2007.)
Assertion
Ref Expression
supsn  |-  ( ( R  Or  A  /\  B  e.  A )  ->  sup ( { B } ,  A ,  R )  =  B )

Proof of Theorem supsn
StepHypRef Expression
1 dfsn2 4040 . . . 4  |-  { B }  =  { B ,  B }
21supeq1i 7908 . . 3  |-  sup ( { B } ,  A ,  R )  =  sup ( { B ,  B } ,  A ,  R )
3 suppr 7930 . . . 4  |-  ( ( R  Or  A  /\  B  e.  A  /\  B  e.  A )  ->  sup ( { B ,  B } ,  A ,  R )  =  if ( B R B ,  B ,  B
) )
433anidm23 1287 . . 3  |-  ( ( R  Or  A  /\  B  e.  A )  ->  sup ( { B ,  B } ,  A ,  R )  =  if ( B R B ,  B ,  B
) )
52, 4syl5eq 2520 . 2  |-  ( ( R  Or  A  /\  B  e.  A )  ->  sup ( { B } ,  A ,  R )  =  if ( B R B ,  B ,  B
) )
6 ifid 3976 . 2  |-  if ( B R B ,  B ,  B )  =  B
75, 6syl6eq 2524 1  |-  ( ( R  Or  A  /\  B  e.  A )  ->  sup ( { B } ,  A ,  R )  =  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   ifcif 3939   {csn 4027   {cpr 4029   class class class wbr 4447    Or wor 4799   supcsup 7901
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-po 4800  df-so 4801  df-iota 5551  df-riota 6246  df-sup 7902
This theorem is referenced by:  supxrmnf  11510  ramz  14405  xpsdsval  20711  ovolctb  21728  nmoo0  25479  nmop0  26678  nmfn0  26679  esumnul  27810  esum0  27811  ovoliunnfl  29909  voliunnfl  29911  volsupnfl  29912  fourierdlem79  31713
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